The present invention relates to data transmission systems which use multidimensional channel codes.
As used herein, the term "multidimensional code" refers to a code in which each codeword is comprised of n coordinates, or components, n&gt;2. For example, each codeword of a four-dimensional code takes the form (.alpha. .beta. .gamma. .delta.), where the components .alpha., .beta., .gamma. and .delta. take on predetermined combinations of values. One particularly advantageous application for multidimensional coding is in the transmission of data over a so-called Gaussian channel--a communications channel in which the transmitted signals are corrupted by Gaussian noise. In such a system, each possible value of an input signal indication, e.g., a binary word comprising a plurality of data bits to be transmitted, is assigned to a different member of a preestablished n-dimensional codeword "alphabet." (In these applications the codewords are also referred to as "data symbols.") As each input word is applied at the transmitting end of the system, the assigned member of the codeword alphabet is determined by table look-up or other means and a signal representing the codeword is applied to the channel. At the other end of the channel, the received, noise-corrupted codeword is decoded in a decoder, or decision-forming circuit. The function of the decoder is to form a (hopefully correct) decision as to what member of the codeword alphabet was actually transmitted by finding the codeword within the alphabet to which the received noise-corrupted codeword is closest in n-space. The principal advantage of using multidimensional codes in such applications is that, as taught by C. E. Shannon in his classic paper "Communication in the Presence of Noise," Proc. IRE, Vol. 37, January, 1949, pp. 10-21, the probability of a decoding error at the receiver decrease as the dimensionality of the codewords increases, given a particular channel and a fixed average power in the transmitted codewords.